Mathematics Capstone Course Parallel Lines Cut By A Transversal
Mathematics Capstone Course Parallel Lines Cut By a Transversal I. UNIT OVERVIEW & PURPOSE: The goal of this unit is for students to understand the angles and the properties related to parallel lines. Students will learn multiple methods for verifying that lines are parallel. They will also understand the relationship of parallel lines to transversal lines. It is important for students to see that mathematical concepts serve as useful means to solving problems that affect our everyday lives. Parallel lines are important to understand not only for a mathematics course, but also in everyday life such as in the design of airports, railways, bridges, buildings and many more geometric components of the real world. In this five-lesson thematic unit, students will engage with the concept of parallel lines cut by a transversal line as they design plans for various elements of a fictitious city. Lessons 1 – 4 can be seen as “building block” lessons through which students acquire and practice the skills needed to create the culminating city project. Lesson 5 is a showcase lesson where students present their projects to the class and recap all that they have learned throughout this thematic unit. Students will complete the city project in groups of three or four. We believe that together, they can negotiate meaning and deepen understanding as they work together to apply geometric concepts to each task. Students will be given several criteria, which must be met in the design of their city, and they will be asked to discuss how geometric methods can assist them in creating optimal design for various elements of the city. For example, students will apply the angle theorems to prove lines parallel, practice geometric proofs, constructions and discover the connections to other topics. At the end of this project, students will be asked to articulate their learning in written and oral summaries of their work to be presented to their classmates in the final class lesson. II. UNIT AUTHORS: Whitney Wall Bortz, Radford University Rachel Hall, Lancaster High School Adam Keith, Gate City High School III. COURSE: Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
Mathematical Modeling: Capstone Course IV. CONTENT STRAND: Geometry V. OBJECTIVES: At the end of this unit: 1. Students will understand the definition of parallel lines; 2. Students will understand what it means for parallel lines to be cut by a transversal line; 3. Students will be able to construct parallel lines; 4. Students will be able to verify that lines are parallel using: a. Algebraic methods; b. Coordinate methods; c. Deductive proofs; 5. Students will make connections between the geometric material included in the lesson and real-life examples of parallel lines; 6. Students will demonstrate knowledge of ratios and proportions and the ability to transfer these previously learned concepts to a real world problem. VI. MATHEMATICS PERFORMANCE EXPECTATION(s): Key Focus: MPE.32 – The student will use the relationships between angles formed by two lines cut by a transversal to a) determine whether two lines are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and c) solve real-world problems involving angles formed when parallel lines are cut by a transversal. Additional: MPE.1 – The student will solve practical problems involving rational numbers (including numbers in scientific notation), percentages, ratios, and proportions. VII. CONTENT: Students will use prior knowledge of parallel lines cut by a transversal and geometric constructions to create plans for the construction of fictitious towns. This project will 2 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
encourage students to work cooperatively and to see how these concepts are used in the real world. Lesson One: Parallel lines and Constructions Students will review the concepts of parallel lines and several angles. Students will use Geogebra or constructions by hand to visualize the properties of parallel lines, transversals and angles formed. This will be an interactive lesson based in the use of technology. Students will be able to manipulate lines and angles so that they learn more about variation and relationships amongst these elements. In this lesson, students will also receive the rubric for the final project so that they are aware of the learning objectives that will be incorporated into the final project and presentation. Key Concepts – constructions, parallel lines, interior angles, alternate interior angles, same side interior angles, exterior angles, alternate exterior angles, vertical angles, and corresponding angles. Lesson Two: Street Design Students will create a street map for their city, and the street map must include parallel lines cut by transversals. They will be required to place various buildings or elements of the city at certain points around the city. These points will be indicated by types of angles. Therefore, students must understand the properties of these angles and how to create and identify them. Key concepts –parallel lines, transversal lines, congruence, alternate interior angles Lesson Three: Designing a Parking Lot Students will design a parking lot for a sports complex in their city. They will incorporate a formerly learned concept of ratios in order to calculate the number of parking spots needed. They will be given an estimated number of people who visit the complex and asked to assume that approximately 1/3 of those people will be driving their own vehicle. Therefore, they should show how they calculate the number of spaces needed. For this lesson, students will use geometric concepts and algebraic methods to create a model of a parking lot and to verify that lines are parallel. They will need to construct parking spaces designated by parallel lines. They will learn how to do so on the coordinate plane. These lines will be transversals intersecting vertical lines in the coordinate plane. 3 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
These will completed as an activity that will be started in class and then finished as homework. The handout/worksheet has a total of 15 questions and 3 extension questions. The first 15 will be marked as either 0 (incomplete), 1 (attempted but incorrect), 2 (approaching target) or 2 (target). Feedback will be provided on top of the mark. The three extension questions are worth 1 extra credit point each Key concepts – construction, parallel lines, perpendicular lines, area of a parallelogram, congruence Lesson Four: Designing an Airport Any thriving city should have an airport in order to promote tourism and business traffic. A safe and operational airport is characterized by design rooted in geometric properties. Students will utilize the properties of parallel lines to create a model of an airport with a runway, taxi routes, and gates. Students will use construction methods learned in lesson two, but they will now also be asked to verify that their lines are parallel using postulates and theorems. Key concepts – parallel lines, transversal lines, construction by hand or by computer program, angles of a circle. Lesson Five: Final project presentation Students will have the final project assignment during the class following lesson four and will use that lesson to get started. They will then have a week to work on the project on their own outside of class. The in-class instructional time will move on to the next scheduled topic in the curriculum. In this lesson, students will present their projects to the class and give feedback to one another (peer assessment). Written reflection: Individually, students will complete a written reflection of their learning through the entire unit, summarizing new and reviewed knowledge. In this reflection, each student must include at least two deductive proofs, demonstrating how they can verify that particular lines in their city are parallel. Oral group presentation: Each group will present their city and its properties to the entire class. This presentation should be done in an electronic presentation format. Each group member should speak during the presentation. The group should engage the class in a mathematical discussion about at least one element of their city. Key concepts – ALL concepts from the unit, critical thinking and communication skills VIII. REFERENCE/RESOURCE MATERIALS: 4 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
SMART Board, Geogebra software, Computer lab, document camera, or LCD projector for modeling, markers, calculators, rulers, compasses, poster boards, and handouts (attached to each lesson). IX. PRIMARY ASSESSMENT STRATEGIES: FORMATIVE METHODS Observation: The teacher will observe students as they work on tasks to see if and where any additional help may be needed. Teacher will monitor, observe, and communicate with students as they work in groups. Class discussion and participation: The teacher will engage students in the discussion during the instructive portion of each lesson, monitoring for understanding. Homework: Homework assignments will be given after each lesson. The homework will be related to the unit plan task and discussed with the class. Handouts will be given to students during class with activities related to the lesson’s learning objectives, and unfinished problems can be done for homework. Doing such activities each day will help both students and the teacher check for understanding throughout the unit. Groups will also be expected to work together outside of class on the final city plan project. Reflective journal: Students record their journals for the last five minutes of class. Entries should include key concepts learned each day as well as any remaining questions that they may want to bring up with the teacher outside of class or in the next class. They may also use this journal to draw connections between content of multiple lessons. SUMMATIVE METHODS Written assessment: Students will be incorporating concepts learned into the final city plan project and presentation. Concepts learned will also appear on our chapter test. Application: Completed project model with all materials. Culminating project presentation: Students will synthesize their learning in a written reflection and an oral presentation. X. EVALUATION CRITERIA: 5 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
Students will be observed as they work during class. Whole class discussions will also help the instructor to determine student knowledge. Presentations by the students and feedback from their peers will also serve as an evaluation tool. The final assessment will be the completed project but all geometric concepts learned will also appear on the written chapter test. XI. INSTRUCTIONAL TIME: We assume block scheduling (90 minute class sessions) This entire unit should be completed in six class sessions. There will be one class session between lessons four and five reserved for groups to work together on the completion of their city projects. 6 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
Lesson 1: Parallel lines and Constructions Strand: Geometry Mathematical Objective(s) Students will be able to Use construction tools to construct parallel lines and transversals. Create parallel lines, transversals, and different angles and visually see how the different angle postulates are represented. Define the following: o Interior angles o Alternate interior angles o Same side interior angles o Exterior angles o Alternate exterior angles o Vertical angles o Corresponding angles Mathematics Performance Expectation(s) MPE 32.a The student will use the relationships between angles formed by two lines cut by a transversal to determine whether two lines are parallel. Virginia SOL G. 2a (The student will use the relationships between angles formed by two lines cut by a transversal to determine whether two lines are parallel.) NCTM Standards Mathematics as Problem Solving Students will demonstrate the ability to use problem-solving approaches to investigate parallel lines and transversals and the angles made by them. Mathematics as Communication Students will communicate mathematical ideas about angle relationships made by parallel lines and transversals. The final lab report gives the opportunity for students to reflect and clarify what they have learned. Mathematics as Reasoning Students will reinforce logical reasoning skills by comparing and contrasting different angle relationships. Mathematical Connections Students will use and value connections between mathematical topics and other disciplines. 7 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
Materials/Resources Compass straight edge paper Geogebra Computer lab Construction instructions – to be used with the constructions using paper and pencil. Smart Camera Smart projector Construction steps if using Geogebra : reproducible at http://www.geogebra.org/book/intro-en.pdf Vocabulary worksheet Final project and rubric Assumption of Prior Knowledge This is a construction exercise that will be built upon in later projects, therefore, the following is important to know and be able to build upon: Students should be able to construct a pair of parallel lines. Students should be able to measure angles. Introduction: Setting Up the Mathematical Task In this lesson, our goal is to have students use Geogebra or construction tools to visualize the properties of parallel lines and transversals in terms of the angles. The goal is to provide an interactive, technology based activity that allows students to manipulate angles and lines more in depth than otherwise possible. By manipulating the lines, students can see what will happen to the angles helping them to see their relationships. Student/Teacher Actions Introduction: Review construction of parallel lines and measuring angles. A. Pass out tools needed for the project. B. Have students construct parallel lines with a transversal and measure the angles made. Teacher will verify that the students can read the protractor and measure angles correctly. C. Explain to the class they will be working on constructing parallel lines cut by a transversal. Construction: Use either Geogebra or constructions tools to do the following: A. Have the students construct parallel lines and a transversal. (Students will be given a clean sheet of paper with the hope that the angle measurements when compared will be different measurements. This will prove it works for any measured angle.) B. Teacher will also construct the lines using the Smart Camera and Smart Projector. C. Have the students mark the angles 1-8. 8 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
D. Using the protractor, students will measure all of the 8 angles made by the parallel lines and transversals. Group discovery: Have the students get into groups to compare their constructions by asking the following questions: (see handout) A. What relationship is there among the interior angles? 1. The two interior angles that line on the same side of the transversal sum to 180 . (They are supplementary) 2. The interior angles that are on opposite sides of the transversal and do not form a linear pair are the same measure. (They are the alternate interior angles ) B. What relationship is there among the exterior angles? 1. The two exterior angles that line on the same side of the straight line sum to 180 degrees. (Same side exterior) 2. The two exterior angles on opposite sides of the straight line are congruent. (They are alternate exterior angles) C. Teacher should use vocabulary to have the students come up with the names of the angles during discussion. This helps students see where the names came from.(Alternate interior, alternate exterior, same side interior, same side exterior) Homework and Extensions Vocabulary worksheet is included to check for understanding and to be used with future assignments. The final product should be presented in an electronic presentation. During the next 5 days, time will be needed in the computer lab or at home to start working on the PowerPoint. (See handout and rubric) Note: Teacher should assign or allow students to choose groups for final project during lesson one. Strategies for Differentiation A. Some students have difficulty using a compass and protractor. Students can be paired with stronger students who will help, or a worksheet with the correct construction may be used to trace. Also, students may use folding techniques to accomplish the same objective. For example: fold a piece of regular sized paper in half and in half again. Then open it to fold it from one corner to the other. This will give 3 parallel lines with one transversal. (See below) 9 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
B. Vocabulary may be difficult. After the discovery is through, have a handout with the vocabulary and angles listed. (See extension vocabulary worksheet) 10 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
Vocabulary for angles made by transversals. Use the figure to identify the following angles: 4 5 8 1. 2. 3. 4. 5. 2 1 3 6 7 Alternate interior angles Alternate exterior angles Same Side interior angles Same Side exterior angles Corresponding angles Which angles are congruent? Which angles are supplementary? 11 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
Group Discovery Handout A. What relationship is there among the interior angles? 1. The two interior angles that line on the same side of the transversal sum to 180 . (They are supplementary) 2. The interior angles that are on opposite sides of the transversal and do not form a linear pair are the same measure. (They are the alternate interior angles ) B. What relationship is there among the exterior angles? 1. The two exterior angles that line on the same side of the straight line sum to 180 degrees. (Same side exterior) 2. The two exterior angles on opposite sides of the straight line are congruent. (They are alternate exterior angles) 12 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
The following instructions were found at m . Begin with point P and line k. 2. Draw an arbitrary line through point P, intersecting line k. Call the intersection point Q. Now the task is to construct an angle with vertex P, congruent to the angle of intersection. 3. Center the compass at point Q and draw an arc intersecting both lines. Without changing the radius of the compass, center it at point P and draw another arc. 4. Set the compass radius to the distance between the two intersection points of the first arc. Now center the compass at the point where the second arc intersects line PQ. Mark the arc intersection point R. 5. Line PR is parallel to line k. 13 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
Handout: CITY PROJECT In this group project, your group will have the opportunity to create a fictitious city. You will be working on various components of your city over the next three lessons. We will have one class period when you can work on the assignment together in class, but you will also need to work together to complete the project outside of class. The final product should be presented in an electronic presentation, such as PPT or Prezi. Your presentation may include text, pictures of example city layouts, and pictures or models of what you have created for your city. If you are unable to put everything into the presentation, it is okay to supply some things in written form or to write things on the board. The following components should be included in your final project: I. II. III. IV. V. VI. VII. Introduction a. Name of city b. Description of city Street Design a. A layout of the streets b. Indication of the properties of the lines and angles c. Contains the following buildings: Parking Lot a. What is the parking lot for? b. How many spaces will there be in the parking lot and how did you decide on the number of spaces? (show your calculations) c. A mini-model of one or two rows in the lot d. A description of how the coordinate plane helped plan the design of these spaces/rows Airport a. Label the runways correctly b. How many terminals will you use? Overall presentation a. Clear communication b. Visuals are good quality and easy to understand c. The group engages the rest of the class in mathematical discussion by asking questions and giving feedback to students on their answers Mathematical Criteria (to be covered at least once at some point in the final project) a. Demonstration or description of how you used a construction b. Demonstration or description of how you used the coordinate plane c. Demonstration or description of how proved lines were parallel in lesson 4. Individual Written reflection (30 points) a. Each student will write a 1-2 page reflection answering the following questions: 1. What did you learn in this unit? Explain the key geometric properties you worked with. (10 points) 2. How did you see geometry applied to the real world in this unit? (5 points) 3. Can you name some real world situations, other than planning a city, where geometry might be needed? Please name at least five. (5 points) 4. What did you like about working in groups? (5 points) 5. What was difficult about working in groups? (5 points) 14 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
Grading Rubric for Reflection Name: Criterion Question 1 Score / 10 Question 2 Score /5 Question 3 Score /5 Question 4 Score /5 Question 5 Score /5 Total Standard 10 – clearly addresses the topic and responds effectively to all aspects of the assignment; 8 – clearly address the topic, but may respond to some aspects of the assignment more effectively than others 6 – addresses the topic, but may slight some aspects of the topic 4 – indicates confusion about the topic or neglects important aspects of the assignment 2 – suggests an inability to comprehend the assignment or to respond meaningfully to the topic 5 – explores the issues showing thorough comprehension of the text; goes beyond the obvious or class discussion 4 – shows some depth and complexity of thought 2 – Has an idea, but gives answers that does not apply 0 – Does not have an answer 5 – explores the issues showing thorough comprehension of the text; goes beyond the obvious or class discussion 4 – shows some depth and complexity of thought 2 – Has an idea, but gives answers that does not apply 0 – Does not have an answer 5 – Gives a detailed list of the good things about working in a group 3 – Addressed the question but does not explain why it is good 0 – No opinion 5 – Gives a detailed list of the bad things about working in a group and how to fix it 3 – Addressed the question but does not explain why it was difficult 0 – No opinion Score /30 15 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
Scoring Rubric for City Project Team Members: 0 1 2 3 Not included Contains some but not all of these items Contains all of these items, but description or rationale is weak Contains all three and provides detailed description and rationale Not included Meets some criteria Meets most criteria Meets all criteria Not included Meets some criteria Meets most criteria Meets all criteria Not included Meets some criteria Meets most criteria Meets all criteria Other element of the city Comments: Not included Meets some criteria Meets most criteria Meets all criteria Overall presentation Comments: Not included Meets some criteria Meets most criteria Meets all criteria Mathematical Not included Criteria Comments: Meets some criteria Meets most criteria Meets all criteria Title, description of city, rationale Comments: Street design Comments: Parking lot Peer Score Teacher Score Comments: Airport Comments: Overall comments: 16 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
Lesson 2: Parallel Lines Project Strand: Geometry Mathematical Objective(s) 1. Students will demonstrate their knowledge of parallel lines with a transversal. 2. Students will show when angles are congruent or supplementary given parallel lines and a transversal. Mathematics Performance Expectation(s) MPE 32.a The student will use the relationships between angles formed by two lines cut by a transversal to determine whether two lines are parallel. Virginia SOL G. 2a (The student will use the relationships between angles formed by two lines cut by a transversal to determine whether two lines are parallel.) NCTM Standards Mathematics as Problem Solving Students will demonstrate the ability to use problem-solving approaches to investigate parallel lines and transversals and the angles made by them. Mathematics as Communication Students will communicate mathematical ideas about angle relationships made by parallel lines and transversals. The final lab report gives the opportunity for students to reflect and clarify what they have learned. Mathematics as Reasoning Students will reinforce logical reasoning skills by comparing and contrasting different angle relationships. Mathematical Connections Students will use and value connections between mathematical topics and other disciplines. Materials/Resources Pencil Colored pencils or markers Ruler Paper (graph paper, if desired) 17 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
Assumption of Prior Knowledge This is a construction exercise that will be built upon in later projects, therefore, the following is helpful to know and be able to build upon although it can be learned quickly: Students should be able to construct a pair of parallel lines. Students should be able to measure angles with a protractor. Introduction: Setting Up the Mathematical Task This activity would be good for students in a Capstone or Geometry class because they are basically just making a map of a city, which would most likely be fun for them. But while they are doing this, they are also thinking about where each building should be. They need to know which angles are congruent and supplementary since the instructions of the placement of the buildings use these terms. Number 4 under instructions is where they really have to think. They have to realize that in order to have non-congruent alternate interior angles; they must turn the map and use the transversals as their 2 lines and one of the parallel lines as their transversal. Overview For this project, each group of 2 students will make a street map for a fictional city (you must name your city). This city will consist of: 1. Six (6) streets that are parallel to each other. Each street should be constructed and named for reference. 2. Two (2) transversal streets. (i.e., Two or more streets that intersect all six of the above streets). These should be named as well. Do not make the transversals parallel to each other; however, one may be perpendicular to the parallel lines!!! 3. Traffic lights or stop signs at four (4) different intersections. 4. The following buildings, represented in your city: a. Post office b. Bank c. Fire Department d. Police Station e. Gas Station f. School g. Restaurant h. Grocery Store i. Sports Complex 18 Developed by Dr. Agida Manizade & Dr. Laura Jacobsen, Radford University MSP project in collaboration with Mr. Michael Bolling, Virginia Department of Education
j. Your own house Instructions (see handout) The point of this project is not to place these buildings anywhere, but to demonstrate your understanding of different angles, as well as to understand when they are supplementary or congruent. You can still be creative in doing so, but please place the buildings in the following locations. 1. 2. 3. 4. 5. Your house and the school at congruent alternate interior angles. The post office and the bank at same side interior angles. The fire department and police station at congruent alternate exterior angles. The restaurant and sports complex at non-congruent alternate interior angles. The gas station and grocery store at congruent corresponding angles. Remember to be creative. You may be
Homework: Homework assignments will be given after each lesson. The homework will be related to the unit plan task and discussed with the class. Handouts will be given to students during class with activities related to the lesson's learning objectives, and unfinished problems can be done for homework. Doing such activities each day will help
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3 Parallel and Perpendicular Lines Mathematical Thinking: Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. 3.1 Pairs of Lines and Angles 3.2 Parallel Lines and Transversals 3.3 Proofs with Parallel Lines 3.4 Proofs with Perpendicular Lines 3.5 Slopes of Lines 3.6 Equations of Parallel and .
1. Lines that do not intersect are parallel lines. 2. Skew lines are coplanar. 3. Transversal is a line that intersects two or more lines. 4. Perpendicular lines are intersecting lines. 5. If two lines are parallel to a third line, then the two lines are parallel. You have just tried describing parallel and perpendicular lines. In
3 Parallel and Perpendicular Lines 3.1 Pairs of Lines and Angles 3.2 Parallel Lines and Transversals 3.3 Proofs with Parallel Lines 3.4 Proofs with Perpendicular Lines 3.5 Equations of Parallel and Perpendicular Lines Tree House (p. 130) Kiteboarding (p. 143) Crosswalk (p. 154) Bike Path (p. 161) Gymnastics (p. 130) Bi
perpendicular lines. . .And Why To write an equation that models part of a leaded glass window, as in Example 6 3-7 11 Slope and Parallel Lines Key Concepts Summary Slopes of Parallel Lines If two nonvertical lines are parallel, their slopes are equal. If the slopes of two distinct nonvertical lines are equal, the lines are parallel.
3-7 Slopes of Parallel and Perpendicular Lines Parallel lines are lines in the same plane that never intersect. All vertical lines are parallel. Non-vertical lines that are parallel are precisely those that have the same slope and different y-intercepts. The graphs below are for the linear equations y 2x 5 and y 2x – 3.
126 Chapter 3 Parallel and Perpendicular Lines 3.1 Lesson WWhat You Will Learnhat You Will Learn Identify lines and planes. Identify parallel and perpendicular lines. Identify pairs of angles formed by transversals. Identifying Lines and Planes parallel lines, p. 126 skew lines, p. 126 parallel
Look at each group of lines. Trace over any parallel lines with a coloured pencil: Lines, angles and shapes – parallel and perpendicular lines 1 2 3 Parallel lines are always the same distance away from each other at any point and can never meet. They can be any length and go in any direc on. ab c ab c Perpendicular lines meet at right angles.
